Prepare the Dressing: In a medium bowl, whisk together the mayonnaise, olive oil, white wine vinegar, grated Parmesan cheese, sugar, oregano, garlic powder, salt, and pepper until well combined and smooth. Taste and adjust seasonings if needed. Set aside.
Assemble the Salad Base: In a large salad bowl, combine the chopped iceberg lettuce and romaine lettuce. Add the halved cherry tomatoes, Kalamata olives, thinly sliced red onion, and sliced pepperoncini peppers.
Dress the Salad: Pour about half of the prepared dressing over the salad ingredients. Gently toss to coat the greens and vegetables evenly. Add more dressing as needed, to your preference. Be careful not to overdress.
Finish and Serve: Sprinkle the shredded Parmesan cheese over the dressed salad. Toss lightly one final time. Serve immediately with extra dressing on the side.